Theory · 3 methods

Electrostatics

What CAPS computes, as it computes it: the equation, its symbols with CAPS's defaults, when to use the method, the source file, the tests that check it, and any departure from the cited method.

Damped shifted force (DSF)

A pairwise alternative to Ewald sums: the Coulomb interaction is damped by erfc and shifted so that both the energy and the force go to zero at the cut-off. It converges to Ewald results for condensed phases at modest cost.

$$E = q_i q_j\left[\frac{\operatorname{erfc}(\alpha r)}{r} - \frac{\operatorname{erfc}(\alpha r_c)}{r_c} + \left(\frac{\operatorname{erfc}(\alpha r_c)}{r_c^2} + \frac{2\alpha}{\sqrt{\pi}}\,\frac{e^{-\alpha^2 r_c^2}}{r_c}\right)(r - r_c)\right], \qquad r < r_c$$

Fennell & Gezelter 2006

Symbol	Meaning	In CAPS
α	damping parameter	0.2 Å⁻¹
r_c	cut-off, shared with van der Waals	10 Å (set per run)
erfc	complementary error function	Abramowitz–Stegun 7.1.26, as LAMMPS

When to use it

The default electrostatics in CAPS: cheap, no grid, fine for neutral bulk polymers. Use smooth PME for ionic systems, interfaces with net dipoles, or when comparing with Ewald-based codes.

Source: core/src/field.cpp
Tested by: Field.ForcesMatchFiniteDifferences · bench/ff/check_data_lammps.py (LAMMPS lj/cut/coul/dsf)
Departure from the reference: bonded partners are treated as LAMMPS coul/dsf does (the damped sum includes them, the excluded part is removed)

References

Fennell, C. J., Gezelter, J. D., "Is the Ewald summation still necessary? Pairwise alternatives to the accepted standard for long-range electrostatics", J. Chem. Phys. 124, 234104 (2006). doi:10.1063/1.2206581

Smooth particle-mesh Ewald (SPME)

The Ewald sum split into a short-ranged real-space part and a smooth reciprocal part evaluated on a grid: charges are spread with B-splines, transformed by FFT, multiplied by the influence function and gathered back as forces.

$$E_\mathrm{rec} = \frac{1}{2\pi V}\sum_{m \ne 0}\frac{\exp(-\pi^2 m^2/\beta^2)}{m^2}\,B(m)\,\left|F(Q)(m)\right|^2, \qquad \operatorname{erfc}(\beta r_c) = \epsilon_\mathrm{rel}$$

Essmann et al. 1995

Symbol	Meaning	In CAPS
β	Ewald coefficient	from erfc(β r_c) = 10⁻⁵, as GROMACS
p	B-spline order	5
h	largest grid spacing	1.0 Å
Q	charge grid	—

When to use it

Charged or strongly polar systems, ionic crystals, interfaces, and whenever results must match Ewald-based codes. Periodic cells only; about 1.5 × the cost of DSF in MD.

Source: core/fortran/caps_kspace.f90 · core/src/kspace.cpp
Tested by: Kspace.PmeMatchesEwaldForForcesEnergyAndVirial · Kspace.MadelungConstantOfRockSalt · Kspace.FftMatchesTheDirectTransform · bench/ff/check_data_lammps.py --pme
Departure from the reference: none

References

Essmann, U., Perera, L., Berkowitz, M. L., Darden, T., Lee, H., Pedersen, L. G., "A smooth particle mesh Ewald method", J. Chem. Phys. 103, 8577–8593 (1995). doi:10.1063/1.470117

Ewald summation (reference)

The plain Ewald sum over reciprocal-lattice vectors. CAPS keeps it as the reference the PME kernel is tested against; runs use PME.

$$E = E_\mathrm{real} + E_\mathrm{rec} + E_\mathrm{self}, \qquad E_\mathrm{self} = -\frac{\beta}{\sqrt{\pi}}\sum_i q_i^2$$

Ewald 1921

Symbol	Meaning	In CAPS
β	Ewald coefficient	as PME
k_max	reciprocal vectors	enough for 10⁻⁸ relative accuracy in tests

When to use it

Not offered for runs: its cost grows as N^{3/2}–N². It checks PME forces, energy and virial.

Source: core/fortran/caps_kspace.f90
Tested by: Kspace.PmeMatchesEwaldForForcesEnergyAndVirial
Departure from the reference: reference only

References

Ewald, P. P., "Die Berechnung optischer und elektrostatischer Gitterpotentiale", Ann. Phys. 369, 253–287 (1921). doi:10.1002/andp.19213690304